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Theorem mulsrmo 7193
Description: There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
Assertion
Ref Expression
mulsrmo ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
Distinct variable groups:   𝑡,𝐴,𝑢,𝑣,𝑤,𝑧   𝑡,𝐵,𝑢,𝑣,𝑤,𝑧

Proof of Theorem mulsrmo
Dummy variables 𝑓 𝑔 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrer 7184 . . . . . . . . . . . . . . . 16 ~R Er (P × P)
21a1i 9 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → ~R Er (P × P))
3 prsrlem1 7191 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → ((((𝑤P𝑣P) ∧ (𝑠P𝑓P)) ∧ ((𝑢P𝑡P) ∧ (𝑔PP))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ) = (𝑡 +P 𝑔))))
4 mulcmpblnr 7190 . . . . . . . . . . . . . . . . 17 ((((𝑤P𝑣P) ∧ (𝑠P𝑓P)) ∧ ((𝑢P𝑡P) ∧ (𝑔PP))) → (((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ) = (𝑡 +P 𝑔)) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩))
54imp 122 . . . . . . . . . . . . . . . 16 (((((𝑤P𝑣P) ∧ (𝑠P𝑓P)) ∧ ((𝑢P𝑡P) ∧ (𝑔PP))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ) = (𝑡 +P 𝑔))) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩)
63, 5syl 14 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ ~R ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩)
72, 6erthi 6268 . . . . . . . . . . . . . 14 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )
87adantrlr 469 . . . . . . . . . . . . 13 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )
98adantrrr 471 . . . . . . . . . . . 12 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )
10 simprlr 505 . . . . . . . . . . . 12 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )
11 simprrr 507 . . . . . . . . . . . 12 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )
129, 10, 113eqtr4d 2125 . . . . . . . . . . 11 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))) → 𝑧 = 𝑞)
1312expr 367 . . . . . . . . . 10 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
1413exlimdvv 1820 . . . . . . . . 9 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (∃𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
1514exlimdvv 1820 . . . . . . . 8 (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞))
1615ex 113 . . . . . . 7 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
1716exlimdvv 1820 . . . . . 6 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (∃𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
1817exlimdvv 1820 . . . . 5 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) → (∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ) → 𝑧 = 𝑞)))
1918impd 251 . . . 4 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
2019alrimivv 1798 . . 3 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
21 opeq12 3598 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
2221eceq1d 6258 . . . . . . . . . 10 ((𝑤 = 𝑠𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝑠, 𝑓⟩] ~R )
2322eqeq2d 2094 . . . . . . . . 9 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐴 = [⟨𝑠, 𝑓⟩] ~R ))
2423anbi1d 453 . . . . . . . 8 ((𝑤 = 𝑠𝑣 = 𝑓) → ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R )))
25 simpl 107 . . . . . . . . . . . . 13 ((𝑤 = 𝑠𝑣 = 𝑓) → 𝑤 = 𝑠)
2625oveq1d 5606 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑤 ·P 𝑢) = (𝑠 ·P 𝑢))
27 simpr 108 . . . . . . . . . . . . 13 ((𝑤 = 𝑠𝑣 = 𝑓) → 𝑣 = 𝑓)
2827oveq1d 5606 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑣 ·P 𝑡) = (𝑓 ·P 𝑡))
2926, 28oveq12d 5609 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → ((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)) = ((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)))
3025oveq1d 5606 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑤 ·P 𝑡) = (𝑠 ·P 𝑡))
3127oveq1d 5606 . . . . . . . . . . . 12 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑣 ·P 𝑢) = (𝑓 ·P 𝑢))
3230, 31oveq12d 5609 . . . . . . . . . . 11 ((𝑤 = 𝑠𝑣 = 𝑓) → ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢)) = ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢)))
3329, 32opeq12d 3604 . . . . . . . . . 10 ((𝑤 = 𝑠𝑣 = 𝑓) → ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩ = ⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩)
3433eceq1d 6258 . . . . . . . . 9 ((𝑤 = 𝑠𝑣 = 𝑓) → [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R )
3534eqeq2d 2094 . . . . . . . 8 ((𝑤 = 𝑠𝑣 = 𝑓) → (𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R ))
3624, 35anbi12d 457 . . . . . . 7 ((𝑤 = 𝑠𝑣 = 𝑓) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R )))
37 opeq12 3598 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ⟩)
3837eceq1d 6258 . . . . . . . . . 10 ((𝑢 = 𝑔𝑡 = ) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝑔, ⟩] ~R )
3938eqeq2d 2094 . . . . . . . . 9 ((𝑢 = 𝑔𝑡 = ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))
4039anbi2d 452 . . . . . . . 8 ((𝑢 = 𝑔𝑡 = ) → ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R )))
41 simpl 107 . . . . . . . . . . . . 13 ((𝑢 = 𝑔𝑡 = ) → 𝑢 = 𝑔)
4241oveq2d 5607 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → (𝑠 ·P 𝑢) = (𝑠 ·P 𝑔))
43 simpr 108 . . . . . . . . . . . . 13 ((𝑢 = 𝑔𝑡 = ) → 𝑡 = )
4443oveq2d 5607 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → (𝑓 ·P 𝑡) = (𝑓 ·P ))
4542, 44oveq12d 5609 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → ((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)) = ((𝑠 ·P 𝑔) +P (𝑓 ·P )))
4643oveq2d 5607 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → (𝑠 ·P 𝑡) = (𝑠 ·P ))
4741oveq2d 5607 . . . . . . . . . . . 12 ((𝑢 = 𝑔𝑡 = ) → (𝑓 ·P 𝑢) = (𝑓 ·P 𝑔))
4846, 47oveq12d 5609 . . . . . . . . . . 11 ((𝑢 = 𝑔𝑡 = ) → ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢)) = ((𝑠 ·P ) +P (𝑓 ·P 𝑔)))
4945, 48opeq12d 3604 . . . . . . . . . 10 ((𝑢 = 𝑔𝑡 = ) → ⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩ = ⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩)
5049eceq1d 6258 . . . . . . . . 9 ((𝑢 = 𝑔𝑡 = ) → [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )
5150eqeq2d 2094 . . . . . . . 8 ((𝑢 = 𝑔𝑡 = ) → (𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))
5240, 51anbi12d 457 . . . . . . 7 ((𝑢 = 𝑔𝑡 = ) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑢) +P (𝑓 ·P 𝑡)), ((𝑠 ·P 𝑡) +P (𝑓 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )))
5336, 52cbvex4v 1848 . . . . . 6 (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R ))
5453anbi2i 445 . . . . 5 ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) ↔ (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )))
5554imbi1i 236 . . . 4 (((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞) ↔ ((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
56552albii 1401 . . 3 (∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞) ↔ ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑠𝑓𝑔((𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ) ∧ 𝑞 = [⟨((𝑠 ·P 𝑔) +P (𝑓 ·P )), ((𝑠 ·P ) +P (𝑓 ·P 𝑔))⟩] ~R )) → 𝑧 = 𝑞))
5720, 56sylibr 132 . 2 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞))
58 eqeq1 2089 . . . . 5 (𝑧 = 𝑞 → (𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
5958anbi2d 452 . . . 4 (𝑧 = 𝑞 → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
60594exbidv 1793 . . 3 (𝑧 = 𝑞 → (∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )))
6160mo4 2004 . 2 (∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ↔ ∀𝑧𝑞((∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ) ∧ ∃𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R )) → 𝑧 = 𝑞))
6257, 61sylibr 132 1 ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283   = wceq 1285  wex 1422  wcel 1434  ∃*wmo 1944  cop 3425   class class class wbr 3811   × cxp 4399  (class class class)co 5591   Er wer 6219  [cec 6220   / cqs 6221  Pcnp 6753   +P cpp 6755   ·P cmp 6756   ~R cer 6758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-iplp 6930  df-imp 6931  df-enr 7175
This theorem is referenced by:  mulsrpr  7195
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